1. Warm-Up Determine the coordinates of each point in the graph below. y
-12
-10 -8 -6 -4 -2 2 4 6 8 10 x y A B C D

2. Linear Equations and Their Graphs pt. 1
Objectives:
To determine whether an ordered pair is a solution of an equation

3. Solutions of Equations
How many solutions does the equation 4x + 6 = 14 have?
What are they?
4x = 8 4 4
x = 2
An equation such as y = 3x + 7 has many solutions, which we write as ordered pairs of numbers.
(x,y)

4. Example 1 Determine whether is a solution of y = 3x -2. ( , ) 2 4
( , ) 2 4
y = 3x - 2
(4)
= 3
(2)
- 2
4 = 6 - 2
4 = 4
(2,4) is a solution of y = 3x -2

5. Practice 1) Determine whether (2,3) is a solution of y = 2x + 3.

6. Example 2 Find three solutions of y = 2x + 11. x y y = 2x + 1111
y = 2(-1) + 11
y = 13
y = 11
y = 9 1 13 -1 9
*Our three solutions are (0,11), (1,13), and (-1,9).

7. Linear Equations and Their Graphs: pt. 2
Objective:
To graph equations in two variables
Using 3 Points
Using Intercepts
Horizontal / Vertical

8. Linear Equations
Equations whose graphs are lines are linear equations.
Here are some examples:
Linear
Equations
y = 3x + 7
6y = -2
9x – 15y = 7
Nonlinear
Equations
y = x2 - 4
x2 + y2 = 16
xy = 3

9. Example 1 Graph the equation 2x + 2y = 6 using 3 points 2x + 2y = 6
solve for y
2x + 2y = 6
-2x
-2x x y
2y = 6 – 2x 3 2 2 1 2
y = 3 - x -2 5
y = 3 - (0)
= 3
y = 3 - (1)
= 2
y = 3 - (-2)
= 5

10. Example 1 Graph the equation 2x + 2y = 6. x y 3 1 2 -2 5 2x + 2y = 6 84 x y 2 3 -8 -6 -4 -2 2 4 6 8 1 2 -2 -2 5 -4 -6 -8

11. Example 2 Graph the equation 3y – 6 = 9x. 3y – 6 = 9x solve for y +62 3 3 1 5
y = 3x + 2 -2 -4
y = 3(0) + 2
= 2
y = 3(1) + 2
= 5
y = 3(-2) + 2
= -4

12. Example 2 Graph the equation 3y – 6 = 9x. x y 2 1 5 -2 -4 3y – 6 = 9x8
3y – 6 = 9x 6 4 x y 2 2 -8 -6 -4 -2 2 4 6 8 1 5 -2 -2 -4 -4 -6 -8

13. Practice Graph these linear equations using three points.
1) 6x – 2y = -2
2) -10x – 2y = 8

14. Graphing Using Intercepts
The x-intercept of a line is the x-coordinate of the point where the line intercepts the x-axis.
The line shown intercepts the x-axis at (2,0). 8 6 4
We say that the x-intercept is 2. 2 -8 -6 -4 -2 2 4 6 8 -2 -4 -6 -8

15. Graphing Using Intercepts
The y-intercept of a line is the y-coordinate of the point where the line intercepts the y-axis.
The line shown intercepts the y-axis at (0,-6). 8 6
We say that the y-intercept is -6. 4 2 -8 -6 -4 -2 2 4 6 8 -2 -4 -6 -8

16. Example 1 Graph 4x – 3y = 12 using intercepts.
*To find the y-intercept, let x = 0.
4x – 3y = 12 x y
4(0) – 3y = 12 -4
0 – 3y = 12
-3y = 12 -3 -3
y = -4

17. Example 1 Graph 4x – 3y = 12 using intercepts.
*To find the x-intercept, let y = 0.
4x – 3y = 12 x y
4x – 3(0) = 12 -4
4x - 0 = 12 3
4x = 12 4 4
x = 3

18. Example 1 Graph 4x – 3y = 12 using intercepts. x y -4 3 8 6 4 2 -8 -6-4 -8 -6 -4 -2 2 4 6 8 -2 3 -4 -6 -8

19. Example 2 Graph 2x + 5y = 10 using intercepts.
*To find the y-intercept, let x = 0.
2x + 5y = 10 x y
2(0) + 5y = 10 2
0 + 5y = 10
5y = 10 5 5
y = 2

20. Example 2 Graph 2x + 5y = 10 using intercepts.
*To find the x-intercept, let y = 0.
2x + 5y = 10 x y
2x + 5(0) = 10 2
2x + 0 = 10 5
2x = 10 2 2
x = 5

21. Example 2 Graph 2x + 5y = 10 using intercepts. x y 2 5 8 6 4 2 -8 -62 -8 -6 -4 -2 2 4 6 8 -2 5 -4 -6 -8

22. Practice Graph using intercepts. 1) 5x + 7y = 35 2) 8x + 2y = 24

23. Warm-Up 10 minutes Graph these equations: -x + 2y = 4 2x + 3y = 8

24. Graphing Horizontal and Vertical Lines
The standard form of a linear equation in two variables is Ax + By = C, where A,B, and C are constants and A and B are not both 0.
3x + 4y = 12
6x + 7y = 23

25. Example 1 Graph y = -2. write the equation in standard form
Ax + By = C
(0)x + (1)y = -2 -8 -6 -4 -2 2 4 6 8
for any value of x
y = -2

26. Example 2 Graph x = 7. write the equation in standard form Ax + By = C
(1)x + (0)y = 7 -8 -6 -4 -2 2 4 6 8
x = 7
for any value of y

27. Practice
Graph these equations.
1) x = 5
2) y = -4
3) x = 0